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Metamath Proof Explorer

Theorem ralxfrd

Description: Transfer universal quantification from a variable x to another variable y contained in expression A . (Contributed by NM, 15-Aug-2014) (Proof shortened by Mario Carneiro, 19-Nov-2016) (Proof shortened by JJ, 7-Aug-2021)

Ref Expression
Hypotheses ralxfrd.1 ⊢ φ ∧ y ∈ C → A ∈ B
ralxfrd.2 ⊢ φ ∧ x ∈ B → ∃ y ∈ C x = A
ralxfrd.3 ⊢ φ ∧ x = A → ψ ↔ χ
Assertion ralxfrd ⊢ φ → ∀ x ∈ B ψ ↔ ∀ y ∈ C χ

Proof

Step Hyp Ref Expression
1 ralxfrd.1 ⊢ φ ∧ y ∈ C → A ∈ B
2 ralxfrd.2 ⊢ φ ∧ x ∈ B → ∃ y ∈ C x = A
3 ralxfrd.3 ⊢ φ ∧ x = A → ψ ↔ χ
4 3 adantlr ⊢ φ ∧ y ∈ C ∧ x = A → ψ ↔ χ
5 1 4 rspcdv ⊢ φ ∧ y ∈ C → ∀ x ∈ B ψ → χ
6 5 ralrimdva ⊢ φ → ∀ x ∈ B ψ → ∀ y ∈ C χ
7 r19.29 ⊢ ∀ y ∈ C χ ∧ ∃ y ∈ C x = A → ∃ y ∈ C χ ∧ x = A
8 3 exbiri ⊢ φ → x = A → χ → ψ
9 8 impcomd ⊢ φ → χ ∧ x = A → ψ
10 9 rexlimdvw ⊢ φ → ∃ y ∈ C χ ∧ x = A → ψ
11 7 10 syl5 ⊢ φ → ∀ y ∈ C χ ∧ ∃ y ∈ C x = A → ψ
12 11 adantr ⊢ φ ∧ x ∈ B → ∀ y ∈ C χ ∧ ∃ y ∈ C x = A → ψ
13 2 12 mpan2d ⊢ φ ∧ x ∈ B → ∀ y ∈ C χ → ψ
14 13 ralrimdva ⊢ φ → ∀ y ∈ C χ → ∀ x ∈ B ψ
15 6 14 impbid ⊢ φ → ∀ x ∈ B ψ ↔ ∀ y ∈ C χ